Another month, another update. Today I’ve added one new note on ample families of sheaves and updated two other notes with substantial new content:
- New Notes: Ample Families (AMF) If you grew up on Hartshorne’s book like I did, then you may have only encountered ample sheaves on a noetherian scheme. In fact the original definition in EGA is given for arbitrary schemes, and in the first section we basically just translate EGA II 4.5.2. Then we define an ample family of sheaves following SGA 6.II 2.2.3 and in particular observe that the tensor powers of an ample family generate the category of quasi-coherent sheaves (which in general has a pretty uninspired set of generators, so this is nice). It is shown in SGA (but not in our notes) that any nonsingular variety over a field has an ample family of sheaves, so this notion of an “ample family” is quite useful.
- Derived Categories of Sheaves (DCOS) I’ve fleshed out the section on the derived Hom and Tensor adjunction with some technical details that become crucial later (for example, the adjunction commutes with restriction). There is also an explicit description of what happens to a map under the adjunction, with a very careful treatment of some complicated sign issues.
After working through these details one comes to understand that there is a simple technique for reducing all the complicated statements on the level of the derived adjunction to statements on the level of the adjunction on complexes, which one can check very efficiently using the explicit description alluded to above. Firstly, in every statement about the derived structures one expects a compatibility diagram relating the derived and underived structures. Usually the compatibility morphisms are isomorphisms for hoinjective/hoflat complexes, so one can reduce to these cases and then using the compatibility diagram reduce to the level of complexes.
There is also a detailed study of the units and counits of this adjunction. By playing around with these morphisms one obtains “for free” several useful morphisms, such as the “double dual” morphism, and the “tensor inside Hom” morphism.
- Derived Categories of Quasi-Coherent Sheaves (DCOQS) On the level of sheaves of modules one defines derived inverse image and tensor using hoflat complexes, and to prove there are enough of these one uses the “extension by zero” construction. Unfortunately this produces bad (i.e. non-quasi-coherent) sheaves in general, so there is something of a problem when one comes to define the derived inverse image and tensor product for quasi-coherent sheaves. This is treated in detail in a new section of the notes.
The section on perfect complexes has also been expanded, so that it now includes the proof that on a nice scheme compact = perfect.
Finally we use the study of adjunctions in DCOS to define the derived projection morphism and “friends” such as the “double dual” and “tensor inside Hom” morphisms. Generally these are isomorphisms when enough things are perfect (an odd sentence). Essentially we give a modern treatment of the derived dual as studied in SGA, back in the days when unbounded complexes were more fearsome.
Anything not from SGA is probably from Neeman’s paper on Grothendieck duality, even if individual attributions aren’t yet in the notes.